For Manchester United, the relief was considerable bearing in mind the weight of history that would have come down on them if they had been unable to avenge last week's FA Cup defeat. It is not since the autumn of 1961 that they last suffered the indignity of four successive defeats and, to put it into perspective, that was also the year Bill Nicholson's Tottenham won the Double, the Berlin Wall went up and a new band going by the name of the Beatles played their first gig at the Cavern. David Moyes desperately needed some form of improvement to prevent another unwanted milestone.

They took their time and there were parts of the first half when, once again, this scarcely resembled the team that had swept up all those league championships under his predecessor. Yet they freewheeled to the finishing line once Antonio Valencia and Danny Welbeck scored within 15 minutes of the restart. Adnan Januzaj reiterated his growing importance to the team and Swansea were probably fortunate to avoid a more emphatic defeat.

The paradox is that Michael Laudrup's team had two-thirds of the ball in the first half. Their problem was that they did not show enough ambition with it, and that was strange because they should have known their opponents might be vulnerable.

Instead, it felt like more like the old United in the second half, with Old Trafford finding its voice and sporadic flashes of brilliance from Januzaj, taking over the role as star performer in the absence of Wayne Rooney and Robin van Persie. It might not be ideal that the champions are so reliant on an 18-year-old in his breakthrough season but, in another sense, when he plays with this distinction it is easy to see why his manager and team-mates have placed so much trust in the youngster.

It spoke volumes, for starters, that Moyes gave him the No10 role at first instead of Shinji Kagawa, who had been brought to the club from Borussia Dortmund as a specialist in the position. When they switched in the second half it was so Januzaj could get at Angel Rangel, on the basis that Swansea's right-back had been booked in the opening 45 minutes. Januzaj's responsibility even stretches to free-kicks these days – he curled an early one against the crossbar – and he was often the player who showed the guile and skill traditionally associated with his club.

He was prominently involved in both goals and, more than anything, always wanted the ball whenever possible. The breakthrough came within a minute of the restart when Januzaj crossed from the left, Ashley Williams was unable to get any real distance with his header and the ball skimmed off the Swansea captain to present Kagawa with the first scoring chance. Gerhard Tremmel saved the midfielder's header but the rebound fell kindly for Valencia and the goal was exposed as he turned in his shot.

The goal came at a good time for the home side because it had been another flat 45 minutes from Moyes's team. For the most part, the crowd remained supportive. Yet there were brief pockets of annoyance, such as the groans that accompanied a careless pass from Chris Smalling, hitting the ball long under no pressure what-soever.

Old Trafford has been remarkably patient so far, in keeping with Sir Alex Ferguson's farewell speech against the same opponents here last May, but Moyes badly needed that sharp improvement in performance after the break.

Welbeck really ought to have soothed the crowd's nerves long before that point, making a pig's ear of the chance that came to him via Valencia and Rafael da Silva in United's most illuminating move of the first half. He has, however, been scoring more regularly this season and it was a clever, instinctive touch for his goal.

Again, Januzaj deserves a lot of credit, intercepting a long throw from Tremmel that was intended for Wayne Routledge. Swansea were in trouble from that moment, culminating in Patrice Evra turning the ball into the penalty area and Welbeck flicking out his foot to add a subtle change of direction.

Smalling should have made it 3-0 shortly afterwards, volleying into the Stretford End after a lovely cross from Kagawa, who improved as the game went on and had his own chance to enhance the margin of victory. Leon Britton rescued his team with a goalline clearance and, again, it should not be overlooked that a swift counter-attacking move from one end of the pitch to the other had begun with Januzaj's wonderfully measured pass out of defence.

The teenager had done more than most to disperse some of the disquiet surrounding United before next Sunday's game against Chelsea at Stamford Bridge. He would be an ideal wearer of the No7 shirt that means so much to this club and is currently lying dormant.